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DigSILENT PF - 05 short circuit theory

H
Himmelstern

This document provides an overview of short-circuit calculations, including the following key points: - Short-circuit calculations are used for system planning and operations to ensure equipment ratings are not exceeded and protective devices are properly coordinated. - The time dependence of short-circuit current is important, as it affects equipment loading. Key current parameters analyzed at different time domains are defined. - The symmetrical components method is used to split three-phase systems into positive, negative, and zero sequence networks to simplify analysis. - Short-circuits are classified based on the phases involved: three-phase, phase-to-phase, phase-to-phase-ground, or single-phase ground. - Common system elements

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Update: June 12, 2003 Short-Circuit Calculations Basic Principles and Models Training Course Documents
Contents 1 The Role of Short-Circuit Calculations 1 1.1 Areas of Applications of Short-Circuit Calculations 1 1.2 Time Dependence of Short-Circuit Current 2 1.3 The System of the Symmetrical Components 3 1.4 Short-Circuit Classification According to Involved Phases 5 2 Plant Models 7 2.1 External Grid 7 2.2 Overhead Lines and Cables 7 2.3 Two-Winding Transformer 8 2.4 Three-Winding Transformer 8 2.5 Series Reactance (Short-Circuit Current Limiting Reactor) 9 2.6 Synchronous Machine 9 2.7 Asynchronous Machine 10 2.8 Loads and Static Shunt-Compensators 10 3 Superposition Method for Short-circuit Calculations 11 4 IEC 60909 Method 12 4.1 Derivation of the Method 12 4.2 Correction Factors of IEC 60909 13 4.3 The Changes from IEC909: 1988 to IEC 60909:2001 15 5 Short-Circuit Currents in the Different Time Domains (according to IEC) 17 5.1 Classification of the Source of Short-Circuits 17 5.2 Initial Symmetrical Short-Circuit Current I"k (according to IEC 60909) 18 5.3 Peak Short-Circuit Current (according to IEC 60909) 18 5.4 Decaying (Aperiodic) DC Component of the Short-Circuit Current idc 20 5.5 Symmetrical Short-Circuit Breaking Current Ib 21 5.6 Steady-State Short-Circuit Current Ik 22 5.7 Thermal Equivalent Short-Circuit Current Ith 22 6 Earthing of Distribution Networks 23 7 Appendix 25 7.1 References 25 7.2 Symbols 25 7.3 Hyphens 25 7.4 Indices 26
- 1 - 1 The Role of Short-Circuit Calculations 1.1 Areas of Applications of Short-Circuit Calculations Apart from the load flow calculation, short-circuit analysis is the most frequently used calculation function when dealing with electrical networks. It is used in system planning as well as system operations (see Figure 1.1). Figure 1.1: Areas of application for short-circuit calculations [2] Applications in system planning are for example: • Ensuring that the defined short-circuit capacity of equipment is not exceeded with system expansion and system strengthening. • Co-ordination of protective equipment (Fuses, over-current and distance relays). • Dimensioning of earth mats • Verification of sufficient fault level capacities at load points (e.g. uneven loads like arc furnaces, thyristor-driven variable speed drives or dispersed generation. • Verification of allowed thermal limits of cables and transmission lines. Applications in system operations are for example: • Ensuring that short-circuit limits are not exceeded when changing the system configuration • Determining protective relay settings as well as fuse sizing • Calculation of fault location for protective relays, which store fault disturbance recordings. • Analysis of system faults, e.g. mal-operation of protection equipment. • Analysis of possible mutual interference of parallel lines during system faults. The fundamental difference for the calculation assumptions is that for system planning studies the system operating conditions are not yet known, and therefore estimates are necessary. For this purpose the method of the equivalent voltage source at the fault location has generally become accepted in Western Europe according to IEC 909 (VDE 0102). A revised version of this was published as IEC 60909 in July 2001. This method works independently to the loadflow of a system. It is based on the nominal and/or calculated dimensions of the operating plant of a system and uses correction factors for voltages and impedances, to ‘push’ the results towards the safe side. For the calculation of minimum and maximum short-circuit currents, different correction factors are applied. Operating / Working Conditions Online Short-Circuit Calculations Planning Criteria Simplified procedure (IEC, ANSI, ...) Reduced Data Set Detailed procedures Complete Data Set Method 1: Equivalent Voltage Source at the Fault Location Method 2.1: Beat Method / Superposition Method Method 2.2: Lösung der DGL sub-transient (initial symmetrical) Short-circuit current I k " ip Ib Ith κ µ m, n I"k, Uk i ik(t) Operating Conditions Online short -circuit calculations Planning Conditions Simplified method (IEC, ANSI, ...) Reduced data set Detailed method Complete data set Method 1: Equivalent voltage source at fault location Method 2.1: Superposition method Method 2.2: Solving of differential eq. Initial symmetrical (subtransient) short- circuit current I k " ip Ib Ith κ µ m, n I"k, Uk i ik(t)
- 2 - For short-circuit calculations in a system operation environment the exact network operating conditions are well known. If the accuracy of the calculation according to IEC 60909 is not sufficient - or to verify the results of this method, the superposition method can be used. It calculates the expected short-circuit currents in the network on the basis of the existing network operating condition. If the system models are correct, the results from this method are always more exact than the results of the method according to IEC 60909. The system analyst is, however, responsible that he has chosen the most unfavourable conditions with respect to the sizing of plant. In individual cases, this might result in extensive studies required. 1.2 Time Dependence of Short-Circuit Current The time dependence of short-circuit current is of importance with respect to the loading experienced by affected plant. In principle one distinguishes between a short circuit that is ‘near to’ or ‘far from’ a generator. (Figure 1.2). Top envelope DC-Component idc Time Current Bottom envelope Top envelope DC-Component idc Time Current Bottom envelope a) Far from generator b) Near to generator Figure 1.2: Time dependence of short-circuit current The distinction of the time dependence is due to the effect of the higher stator current (due to the close fault) on the induced currents in the damper windings, rotor mass and field winding. In the case of a fault near to a generator the stator current can increase so much that the resulting magnetic field weakens the rotor field considerably. As a consequence the terminal voltage collapses. The associated positive sequence model of a synchronous machine is shown in Figure 1.3. The delayed effect of the stator field on the excitation (rotor) field is modelled by switching between the source voltage E", E' and E depending on the time frame of the calculation. xd-x'd S x'd-x"d x"d E' E"E ∞ t Figure 1.3: Single-phase equivalent circuit diagram of a generator for short-circuit current calculations which include the modelling of the field attenuation For all practical calculations the IEC60909 standard assumes a fault near to a generator if the fault current in at least one generator exceeds twice the rated current. The short-circuit current is described by the following parameters: ip Peak short-circuit current, i.e. magnitude of the first instantaneous peak value of the short-circuit current. It is used for the calculation of the mechanical load of electrical plant (e.g. busbars in switchgears, transformers). This value depends on the R/X relationship of the fault impedance and the angle of the short-circuit current inception. idc Decaying (aperiodic / direct-current) component of the short-circuit current. This value is of interest to determine the thermal effect of the short-circuit current. In the case of faults near to the generator, it also affects the time up to the first zero crossing of the current.
- 3 - Ik" Initial symmetrical (subtransient) short-circuit current. This is the rms value of the alternating current at the inception of the short-circuit. Ik' Transient short-circuit currentThis is the rms value of the alternating current at the transition from sub-transient short-circuit current to steady state short-circuit current. Ik Steady state short-circuit current. This is the rms value of the alternating current in the steady-state condition. Not possible to determine from the time dependence, but often used as a derived quantity: knk "IU3"S ⋅⋅= Initial symmetrical (subtransient) short-circuit power. This is calculated as the fictitious product of the rated voltage Un (approximated as the voltage before short-circuit inception) and the initial short- circuit current Ik". 1.3 The System of the Symmetrical Components 1.3.1 Concept of Symmetrical Components Using the method of the symmetrical components a three-phase AC system (including the capacitive and inductive couplings) can be split into three independent single phase systems namely Positive, Negative and Zero sequence networks. Three phasors are introduced, which have a magnitude of unity, but are 120° phase displaced: 1, a and a2 . With: 2 3 j 2 1 a +−= (Eq. 1.1) 2 3 j 2 1 a2 −−= (Eq. 1.2) 0aa1 2 =++ (Eq. 1.3) Im Re 1 = a3 a2 a 120° 120° 120° Figure 1.4: Symmetrical system with phasors 1, a and a2 a) I1R ωt I1S I1T b) I1R ωt I1T I1S c) I1R ωt I1T I1S Figure 1.5: Example for Positive, Negative and Zero Sequence currents
- 4 -           ⋅           ⋅=           T S R I I I aa aa I I I 2 2 2 1 0 1 1 111 3 1 (Eq. 1.4) i.e. ( )TSR IIII ++= 3 1 0 ( )TSR IaIaII ⋅+⋅+= 2 1 3 1 ( )TSR IaIaII ⋅+⋅+= 2 2 3 1           ⋅           =           2 1 0 2 2 1 1 111 I I I aa aa I I I T S R (Eq. 1.5) i.e. 210 IIII R ++= 21 2 0 IaIaII S ⋅+⋅+= 2 2 10 IaIaII T ⋅+⋅+= 1.3.2 Determining the Sequence Impedance Values The impedance of electrical plant in the system of the symmetrical components can be determined by switching a voltage supply of the appropriate phase sequence to the terminals of the plant and measuring the resulting currents. The equivalent circuits are shown below. a) b) Figure 1.6: Equivalent circuit to determine the positive sequence impedance (a) and negative sequence impedance (b) Figure 1.7: Equivalent circuit to determine the zero sequence impedance
- 5 - 1.4 Short-Circuit Classification According to Involved Phases Depending on the phases which are involved in a short-circuit, one needs to distinguish between a three-phase short circuit, a phase-phase short circuit, a phase-phase-earth short-circuit or a single phase-earth short-circuit. The different types of short-circuit are shown in Figure 1.8 to 1.11. The illustrations show the system fault in the ABC phase representation (above) and the equivalent circuit in the system of the symmetrical components (below). 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 nU c ⋅ 1I Figure 1.8: Three-phase short-circuit (ABC phase representation and equivalent circuit in symmetrical components) 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 U c n ⋅ 1I 2I Figure 1.9: Phase-to-phase short-circuit (no earth connection) (ABC phase representation and equivalent circuit in symmetrical components)
- 6 - 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 nU c ⋅ 1I 2I 0I Figure 1.10: Phase-phase-earth short-circuit (ABC phase representation and equivalent circuit in symmetrical components) 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 nU c ⋅ 1I 0I 2I Figure 1.11: Single-phase to earth fault (ABC phase representation and equivalent circuit in symmetrical components) The designation earth contact and earth short-circuit depends on whether the currents flowing during an earth fault are short-circuit or load current like. This depends, of course, on the system earthing. One speaks of an earth contact in isolated or resonantly earthed networks, and of earth short-circuits in solidly earthed or resistively earthed systems. In order to calculate the short-circuit current of a network, the positive and negative sequence data for the plant as well as the zero sequence data is required.
- 7 - 2 Element Models For the common element models, the calculated (or actual measured) values are used. The shown models are not complete as required, for example, for loadflow calculation. They rather represent simplifications, which are made in the context of the short-circuit calculations according to the IEC. The detailed equivalent circuit diagrams can be viewed in the PowerFactory manual [5]. 2.1 External Grid ~U11 2 0 RN1 XN1 RN2 XN2 RN0 XN0 Figure 2.1: Short-circuit model for external grid Parameters and calculations: k n 1N "I3 Uc Z ⋅ ⋅ = (Eq. 2.1) Additional data: 1N2N ZZ = RN1, XN1 according to Ratio 1N 1N X R RN0, XN0 according to 1N 0N Z Z und 0N 0N X R 2.2 Overhead Lines and Cables 1 2 0 RL1 XL1 RL2 XL2 RL0 XL0 CL0/2 CL0/2 Figure 2.2: Short-circuit model of lines Parameters and calculations: RL1, XL1 according to conductor geometry or manufacturer data 12 LL ZZ = RL0, XL0 according to conductor/geometry under consideration of additional parallel conductors buried in the ground (e.g. district-heating tubes). Thermal resistance compensation for the calculation of the minimum short-circuit current ( )[ ] 20,LeL RC201R ⋅°−ϑ⋅α+= (Eq. 2.2)
- 8 - 2.3 Two-Winding Transformer 1 2 0 RT,HV1 XT,HV1 RT,HV2 XT,HV2 3ZE1 3ZE2 ZT0 Figure 2.3: Short-circuit model for a two-winding transformer Parameters and calculations: rT 2 HV,rT kr1HV,T S U uZ ⋅= (Eq. 2.3) rT 2 HV,rT Rr1HV,T S U uR ⋅= (Eq. 2.4) 1HV,T2HV,T ZZ = (Eq. 2.5) The zero-sequence equivalent circuit is dependant on the vector group of the transformer (see users manual [5]) 2.4 Three-Winding Transformer 1 0 RT,HV11 XT,HV11 3ZE1 3ZE3 ZT0 RT,HV31 XT,HV21RT,HV21 XT,HV31 2 3ZE2 RT,HV12 XT,HV11 RT,HV32 XT,HV22 RT,HV22 XT,HV32 Figure 2.4: Short-circuit model for a Three-winding transformer Parameters and calculations: 12rT 2 HV,rT 12kr1,12 S U uZ ⋅=Σ (Eq. 2.6) 12rT 2 HV,rT 12Rr1,12 S U uR ⋅=Σ (Eq. 2.7) 23rT 2 HV,rT 23kr1,23 S U uZ ⋅= ∆Σ (Eq. 2.8) 23rT 2 HV,rT 23Rr1,23 S U uR ⋅=Σ (Eq. 2.9) 31rT 2 HV,rT 31kr1,31 S U uZ ⋅=Σ (Eq. 2.10) 31rT 2 HV,rT 31Rr1,31 S U uR ⋅=Σ (Eq. 2.11) ( )1,311,231,1211HV,T ZZZ 2 1 Z ΣΣΣ +−= ( )1,311,231,1221HV,T ZZZ 2 1 Z ΣΣΣ −+= (Eq. 2.12) ( )1,311,231,1231HV,T ZZZ 2 1 Z ΣΣΣ ++−= (Eq. 2.13) 1,HVi,T2,HVi,T ZZ = The zero-sequence equivalent circuit is dependant on the vector group of the transformer (see users manual [5])
- 9 - 2.5 Series Reactance (Short-Circuit Current Limiting Reactor) 1 2 0 RR1 XR1 RR2 XR2 RR0 XR0 Figure 2.5: Short-circuit model for a series reactance Parameters and calculations: rR 2 n kr1R I3 U uZ ⋅ ⋅= (Eq. 2.14) rT 2 rT Rr1R S U uR ⋅= (Eq. 2.15) For a balanced system: 1R0R2R ZZZ == (Eq. 2.16) 2.6 Synchronous Machine ~U"11 2 0 RS1 X"S1 RS2 X"S2 RS0 X"S0 ZE 3ZE S Figure 2.6: Short-circuit model for a synchronous machine Parameters and calculations: dSS "jXRZ += (Eq. 2.17) Additional data: 2S2S2S jXR"jXRZ +=+= (Eq. 2.18) Normally it is assumed that X2 = x"d . If x"d and xq" differ significantly in magnitude, the following can be used: ( )qd xxXX "" 2 1 " 22 +⋅== (Eq. 2.19)
- 10 - 2.7 Asynchronous Machine ASM ~U"11 2 0 RA1 X"A1 RA2 X"A2 RA0 X"A0 Figure 2.7: Short-circuit model for an Asynchronous machine Parameters and calculations: rM 2 rM rM LR AK S U I I 1 Z ⋅       = (Eq. 2.20) 2.8 Loads and Static Shunt-Compensators 1 2 0 RLoad1 XLoad1 not for IEC60909 CLoad1 RLoad2 XLoad2 CLoad2 RLoad0 XLoad0 CLoad0 0 Figure 2.8: Short-circuit model for loads and shunt – compensators The complete equivalent circuit for loads is only used for the superposition method. If IEC60909 is used, the loads are not reflected in the positive and negative sequence networks, but are of importance in the zero-sequence network.
- 11 - 3 Superposition Method for Short-circuit Calculations The superposition method is (in terms of system modelling) an accurate calculation method. The fault currents of the short-circuit are determined by overlaying the healthy loadflow condition before short-circuit inception with a condition where all voltage supplies are set to zero and the negative operating voltage is connected at the fault location. The procedure is shown in Figure 3.1 below. ~ ~ ~ US1 US2 US3 UOp,0 ~ UOp,0 + = UOp,0 ~ ~ ~ US1 US2 US3 USC= 0 IOp IOp IOp ISC ISC ISC ISC + IOp ISC + IOp ISC + IOp a) b) c) Figure 3.1: Superposition method for short-circuit calculations Starting point is the operating condition of the system before short-circuit inception (see Figure 3.1 a). This condition represents the excitation conditions of the generators, the tap positions of regulated transformers and the breaker / switching status of the operational plant. From this pre-fault condition the pre-fault busbar voltage of the faulted busbar can be calculated. For the pure fault condition the system condition is calculated for the situation where, the negative pre-fault busbar voltage for the faulted bus is connected at the fault location and all other sources / generators are set to zero (see Figure 3.1 b). Since network impedances are assumed to be linear, the system condition after fault inception can be determined by overlaying (complex adding) both the pre-fault and pure fault conditions (se Figure 3.1 c).
- 12 - 4 IEC 60909 Method 4.1 Derivation of the Method The method of the equivalent voltage source at the faulted bus is a simplification of the superposition method with the goal of accomplishing a close-to-reality short-circuit calculation without the need for the preceding load flow calculation and the associated definition of actual operating conditions. Figure 4.1 shows, how the method of the equivalent voltage source can be derived from the superposition method. ~ ~ ~ US3=Un3 UOp,0=Un ~ + Un ~ ~ ~ Un1 Un2 Un3 USC= 0 IOp=0 ISC ISC ISC ISC ISC ISC a) b) c) US2=Un2 US1=Un1 IOp=0 IOp=0 c Un ≈ Figure 4.1: Method of the equivalent voltage source at the faulted busbar, derived from the superposition method In comparison, the main simplifications in comparison to the superposition method are the following: • Nominal conditions are assumed for the whole network, i.e. Ui = Un,i. • Load currents are neglected, i.e. IOp,i = 0. • A simplified simulation network is used, i.e. loads are not considered in the positive and negative sequence network. • To ensure that the results are estimated on the safe side, a correction factor c is applied to the voltage at the faulted busbar. This factor differs for the calculation of the maximum and the minimum short-circuit current of a network.
- 13 - 4.2 Correction Factors of IEC 60909 The superposition method always assumes a realistic system condition as the basis for a short-circuit calculation, which is determined from a load flow calculation preceding the short-circuit calculation. Thus the exact short-circuit currents are determined for this system condition. By using the equivalent voltage source in accordance with IEC60909, the aim is to calculate the maximum and minimum short-circuit currents for all possible operating conditions with only one calculation. Only the rated voltage of the faulted bus is required. For this purpose IEC 60909 introduces a voltage correction factor and several impedance correction factors. The concept of impedance correction according to IEC 60909 (see Figure 4.2) is to correct the source impedance in such a way, that in the case of calculating the short-circuit current of a circuit with the voltage c×Un and corrected impedance the same values results as in the case of calculating with the actual fault voltage and actual impedance: ~U"k Zk I"k ~c Un K Zk I"k,IEC Figure 4.2: Concept of impedance correction according to IEC 60909 Thereby the following is applicable: ( ) IEC,k k n k k k "I ZK3 Uc Z3 "U "I = ⋅⋅ ⋅ ≡ ⋅ = (Eq. 4.1) The values for voltage factor c as well as the impedance correction factors K are listed below. 4.2.1 Voltage Factor Table 4.1: Voltage factor c as a function of the nominal voltage Rated / nominal voltage Calculation of max. short- circuit current cmax Calculation of min. short-circuit current Cmin Low Voltage Un ≤ 1 kV 1.05 (with Umax ≤ 1.06 Un) 1.10 (with Umax ≤ 1.10 Un) 0.95 Medium Voltage 1 kV < Un ≤ 35 kV 1.10 1.00 High Voltage 35 kV < Un 1.10 If Un is not defined: cmax⋅Un → Um 1.00 If Un is not defined: cmin⋅Un → 0.9⋅Um In general ensure: cmax ⋅ Un ≤ Um 4.2.2 Impedance Correction for Power Transformers Correction of the transformer impedances in the positive, negative and zero sequence networks (except for earth impedances): krT max T x6.01 c 95.0K ⋅+ ⋅= (Eq. 4.2)
- 14 - If the operating conditions of the transformer prior to the fault inception are known, the following correction factor can be used: max,Tb rT max,Tb rT max max,b n T sin I I x1 c U U K ϕ      ⋅+ ⋅= (Eq. 4.3) Impedance correction in three-winding transformers: 12kr max 12T x6.01 c 95.0K Σ Σ ⋅+ ⋅= (Eq. 4.4) 23kr max 23T x6.01 c 95.0K Σ Σ ⋅+ ⋅= (Eq. 4.5) 31kr max 31T x6.01 c 95.0K Σ Σ ⋅+ ⋅= (Eq. 4.6) 4.2.3 Impedance Correction for Synchronous Machines (Generators) Using IEC 60909, the actual measured resistive portion of the short-circuit impedance Rs may not be used for the real portion of the short-circuit impedance. Instead a fictitious resistance value RSG is introduced, which is significantly higher in comparison to RS and should simulate the decaying DC component. The values of the resistance RSG to be used are shown below. Table 4.2: Choice of the fictitious generator resistance RS/X"d UrG SrG 0.15 ≤ 1kV arbitrary 0.07 > 1kV < 100 MVA 0.05 > 1kV ≥ 100 MVA In addition the positive, negative and zero sequence impedances (with exception of the earthing impedances) of the synchronous machine are to be corrected with the factor KG: rGd max rG n G sin"x1 c U U K ϕ⋅+ ⋅= (Eq. 4.7) 4.2.4 Impedance Correction for Power Stations 4.2.4.1 Power stations with on-load tap changers Impedance correction in the positive, negative and zero sequence network (with exception of the earthing impedances) for power stations: rGTd max 2 r 2 rG 2 Netw,n PS sinx"x1 c t 1 U U K ϕ⋅−+ ⋅⋅= (Eq. 4.8) If the power station is operating under abnormal conditions, (e.g. operating with a voltage at the generator terminals that deviates from UrG, under-excited operation), IEC60909 defines a number of corrections, which “should” be used instead. 4.2.4.2 Power stations with off-load tap changers Impedance correction in the positive, negative and zero sequence network (with exception of the earthing impedances) of the power station: ( ) ( ) rGd max T rGrG Netw,n PS sin"x1 c p1 t 1 p1U U K ϕ⋅+ ⋅+⋅⋅ +⋅ = (Eq. 4.9) If the power station is operating under abnormal conditions, (e.g. operating with a voltage at the generator terminals that deviates from UrG, under-excited operation), IEC60909 defines a number of corrections, which “should” be used instead.
- 15 - 4.2.5 Guidelines for the Modelling of Asynchronous Machines For the real part of the short-circuit impedance - similar to the synchronous machine - reference values are indicated as a function of the rated voltage and power for each pole pair. These are shown in the table below. Table 4.3: Selection of the resistance (according to IEC 0909) RM/XM UrM PrM per pole pair 0.1 > 1kV ≥ 1 MW 0.15 > 1kV < 1 MW 0.42 ≤ 1kV, incl. connection cable arbitrary / random IEC 60909 furthermore defines a number of conditions, under which the fault contribution of asynchronous machines can be neglected. These assumptions are only of meaning in a manual short-circuit calculation. When using a computer simulation program, however, they can be neglected. Therefore reference is only made to section 3.8.2 in [3]. 4.3 The Changes from IEC909: 1988 to IEC 60909:2001 The superposition method is always based on a realistic network condition, which is determined by running a loadflow prior to the actual short-circuit calculation. Thus the exact short-circuit current for this system condition is calculated. The intention of the use of an equivalent voltage source according to IEC60909, is to calculate the maximum and the minimum short-circuit currents for all possible operating conditions with exactly one calculation. In addition only the nominal voltage is to be used at the fault location. The norm therefore introduces a voltage correction factor and several impedance correction factors. 4.3.1 Correction Factors for Power Transformers IEC 909:1988 No impedance correction in principle. Reference is made to special considerations for the following cases: - A single fed short-circuit has the same direction as the load current - Tap changer with an overall voltage ratio > 5% - Voltage Uk,min is significantly lower than the short-circuit voltage Ukr - Operating voltage is significantly higher than Un (U>1.05 Un) IEC60909: 2001 Specific correction factor for power transformers 4.3.2 Correction Factors for Synchronous Generators without Generator Transformers IEC909: 1988 Calculation of the impedance correction with UrG IEC 60909:2001 Substitution of ( ) rGGrG Up0.1U ⋅+→ , if the generator operating voltage continuously deviates from UrG
- 16 - 4.3.3 Correction Factors for Power Stations 4.3.3.1 Factors for power stations with tap change controller IEC909: 1988 Choice between: - Single correction: Impedances of generator and step-up transformer are multiplied by different correction factors. - Total correction: Impedances of generator and step-up transformer are regarded as one impedance and are corrected together. IEC 60909:2001 Only total correction allowed 4.3.3.2 Factors for power stations without tap change controller IEC909: 1988 (No guidelines) IEC 60909:2001 Own correction factor 4.3.4 Maximum Cable Temperature for the Calculation of the Minimum Short-Circuit Current IEC 909:1988 ϑmax = 80 °C IEC60909: 2001 ϑmax corresponds to the maximum possible conductor temperature 4.3.5 Voltage Factor c for Low Voltage Systems 400 / 230V IEC 909:1988 c max = 1.0 IEC60909: 2001 c max = 1.05 (for Ub < 1.06 Un), otherwise cmax = 1.10 4.3.6 Consideration of the Fault Contribution of Asynchronous Machines IEC 909:1988 - consideration of three phase and two phase (to earth) faults - note that for low resistance earthing the single phase to earth fault is also to be considered (no calculation instruction given) IEC60909: 2001 - detailed calculation instruction for all types of fault - concrete rules for the consideration of asynchronous machines (meaningful only for manual calculations) 4.3.7 Standardisation of New Calculation Methods IEC60909: 2001 - calculation instruction for single phase interruption in the medium voltage system (fuse blowing) and faults in the LV system - calculation instruction for the thermal effective short-circuit current as adopted from IEC 865-1
- 17 - 5 Short-Circuit Currents in the Different Time Domains (according to IEC) 5.1 Classification of the Source of Short-Circuits The different types of sources are treated differently during short-circuit current calculations according to IEC 60909. Depending on the type of source, some of the characteristic current values derived from the initial symmetrical short- circuit current I"k are calculated differently. For a single fed source (Figure 5.1) there exists only one path from the feeding voltage source to the fault location. Depending upon the number of such voltage sources one distinguishes between a single source (exactly one voltage source) and a non-meshed source (several voltage sources). For non-meshed sources the portion of short-circuit current that flows in each branch of the network is the current supplied by the one and only source connected to this branch. A meshed source exists (Figure 5.2) if the short-circuit lies in a part of the network, where the portion of short-circuit current in at least one branch of the network is supplied from more than one source. a) b) Figure 5.1: Single fed short-circuits fed from a single source (a) and a Non-meshed network (b) Figure 5.2: Short-circuit is a meshed network
- 18 - 5.2 Initial Symmetrical Short-Circuit Current I"k (according to IEC 60909) 5.2.1 Maximum Initial Symmetrical Short-Circuit Current I"k,max The following steps are performed: - Reproduce the system in symmetrical components, including the indicated impedance corrections. - Insert the equivalent voltage source at the fault location and the calculate k 2 nmax max,k Z 1 3 Uc "I ⋅ ⋅ = (Eq. 5.1) - The currents in the three-phase system are derived from the inverse transformation of the symmetrical components currents. 5.2.2 Minimum Initial Symmetrical Short-Circuit Current I"k,min The procedure is basically the same as the calculation of the maximum initial symmetrical short-circuit current I"k,max. The following differences exist [3]: - The voltage correction is done with the factor cmin instead of cmax. - For overhead lines and cables the resistance values for the maximum possible conductor temperature (instead of 20ºC) should be specified. - The system and supplying generators are to be configured such that the lowest possible short circuit current results. 5.3 Peak Short-Circuit Current (according to IEC 60909) The following steps are performed: - Reproduce the system in symmetrical components, including the indicated impedance corrections. - Insert the equivalent voltage source at the fault location and the calculate kp "I2i ⋅⋅κ= (Eq. 5.2) The calculation of κ is dependant on whether the source is single or meshed. 5.3.1 Calculation of the Factor κ for Unmeshed Systems The value of κ is calculated from the short-circuit impedance Zk = Rk + j Xk as follows: k k X R 3 k k e98.002.1 X R ⋅− ⋅+=      κ=κ (Eq. 5.3) The factor κ is used uniformly for all types of fault.
- 19 - Figure 5.3: Dependence of κ with respect to R/X ratio 5.3.2 Calculation of the Factor κ for Meshed Systems If the decaying direct current component is determined from branches with differing R/X ratios, then the factor κ must be calculated using one of the following methods. The method of equivalent frequency is generally used for computer-aided calculations, as it is simple to implement, yet results in relative high accuracy. 5.3.2.1 Uniform R/X Ratio (Method A) For this method the minimum R/X ratio of all branches through which any portion of the short-circuit current flows is selected. Network plant connected in series will be considered as one branch.       =      i i min X R Min X R i: branches carrying a portion of the short-circuit current (Eq. 5.4)               κ=κ min a X R (Eq. 5.5) This procedure results is a conservative estimate for the value of R/X and therefore leads to larger κ values than in reality. 5.3.2.2 R/X Ratio at the Fault Location (Method B) Here the R/X ratio is determined according to the ratio of the impedance components at the fault location. This generally results in a too small value of κ. Therefore the calculated peak short circuit current needs to be corrected.       κ=κ k k b X R (Eq. 5.6) kbp Ii "215.1 ⋅⋅⋅= κ (Eq. 5.7) The following special rules need to be considered a) If R/X is less than 0.3 for all branches, the 1.15 correction factor need not be applied. b) For low-voltage systems the value of 1.15 x κb should not exceed 1.8. c) For medium / high-voltage transmission systems the value of 1.15 x κb should not exceed 2.0.
- 20 - 5.3.2.3 The Equivalent Frequency Method to Determine the R/X Ratio (Method C) This method consists of three steps: - Adjust the resistances and reactances for all plant components according to Eq. 5.8 and 5.9. The value to use for the equivalent frequency fe is listed in Table 5.1. i * i RR = (Eq. 5.8) e n i * i f f XX ⋅= (Eq. 5.9) - Determine the short-circuit impedance at the fault location for the equivalent frequency fe. - The value of κc is determined from the ratio of the short-circuit impedance components at the equivalent frequency:         κ=κ e e f,k f,k c X R (Eq. 5.10) Table 5.1: Equivalent frequencies for 50 / 60 Hz systems. fn fe 50 Hz 20 Hz 60 Hz 24 Hz 5.4 Decaying (Aperiodic) DC Component of the Short-Circuit Current idc IEC 60909 suggests the following approximation for the time response of the decaying DC component. t X R f2 kdc e"I2i ⋅π− ⋅⋅= (Eq. 5.11) The R/X ratio is determined according to the method of uniform R/X ratio (5.3.2.1) or according to the method of equivalent frequency (5.3.2.3). Note that for synchronous machines the correct values of RS should be used and not the resistances RSf, which are used to determine the time constants of the DC components. For the method of equivalent frequency the equivalent frequency should be chosen according to the following table depending on the time frame studied. Table 5.2: Equivalent frequency fe to use to determine idc for different time frames. f⋅t <1 <2.5 <5 <12.5 fe/ fn 0.27 0.15 0.092 0.055
- 21 - 5.5 Symmetrical Short-Circuit Breaking Current Ib 5.5.1 Unsymmetrical Faults Far from the Generator kb "II = (Eq. 5.12) 5.5.2 Symmetrical Faults Near to the Generator (single source) 5.5.2.1 Fault Contribution from Synchronous Machine / Generator kb "II ⋅µ= (Eq. 5.13) The factor µ(tmin) < 1 depends on the minimum breaker tripping time tmin. It allows for the feedback of the rotor field onto the stator. 5.5.2.2 Fault Contribution from Asynchronous Machine kb "IqI ⋅⋅µ= (Eq. 5.14) The additional factor q (tmin) < 1 depends on the minimum breaker tripping time tmin as well as the rated power per pole pair of the machine. It allows for the decay of the short circuit current contribution of the machine during solid short circuits. 5.5.3 Symmetrical Faults for Meshed Networks 5.5.3.1 Multiple non-meshed sources ∑= i i,bb II (Eq. 5.15) 5.5.3.2 Meshed Sources The total symmetrical short-circuit breaking current can be approximated by adding the breaking currents of each individual source. In addition IEC 60909 provides a formula to improve the accuracy of the result.
- 22 - 5.6 Steady-State Short-Circuit Current Ik 5.6.1 Unsymmetrical Faults Far from the Generator kk "II = (Eq. 5.16) 5.6.2 Symmetrical Faults Near to the Generator (Single Source) S,rmaxmax,k II ⋅λ= (Eq. 5.17) S,rminmin,k II ⋅λ= (Eq. 5.18) The factor λ depends on Xd,sat. IkG"/IrG < 1 depends on the minimum breaker tripping time tmin. It allows for the feedback of the rotor field onto the stator. 5.6.3 Symmetrical Faults Near to the Generator (Unmeshed Source) ∑= i i,kk II (Eq. 5.19) 5.6.4 Symmetrical Faults Near to the Generator (Meshed Source) kk "II = (Eq. 5.20) This rough approximation does not always yield satisfactory results. As an alternative, the short circuit current contributions can be approximated by determining the breaking current Ib (according to IEC 60909) with maximum breaker tripping time. This IEC 60909 compatible solution is implemented in PowerFactory as an option. 5.7 Thermal Equivalent Short-Circuit Current Ith This current is used to verify the thermal capabilities of cables based on the response of the power system protection. ( ) k 2 thk 2 k T 0 2 TITnm"Idti k ⋅=⋅+⋅=∫ (Eq. 5.21) k Tk 0 2 th T dti I ∫ = (Eq. 5.22) m: thermal contribution of the DC component n: thermal contribution of the AC component
- 23 - 6 Earthing of Distribution Networks In this chapter the effect of network earthing on the fault current for a single-phase to earth short circuit is discussed. This application is particularly interesting as it allows, apart from the application for short circuit calculation methods, a general discussion about the different possibilities of earthing in a network. In Figure 6.1 the system to be studied is shown as a single line diagram as well as in the system of symmetrical components. The system model consists of an external grid of the HV transmission system, of a HV-MV step-down transformer with the possibility to earth the starpoint on the MV side via different earth impedances and a line, which represent the MV network. At the end of the line a single-phase to earth fault is placed. For a single-phase to earth fault the positive, negative and zero sequence networks have to be switched in series. The model of the Dy transformer in the zero sequence network exists of an isolated HV side, with the MV shorted to earth via ZT0 and 3ZE. ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 ZE 3 ZE 1PE CL/2CL/2 Figure 6.1: Single phase-to-earth short-circuit represented as a single line diagram and an equivalent symmetrical components network Figure 6.2 shows the equivalent circuit diagrams for a system with a solidly earthed starpoint (or a low resistance current limiting starpoint) (Figure 6.2a), for an isolated starpoint (Figure 6.2b) and for a resonantly earthed (compensated) system (Figure 6.2c). In the case of the solidly earthed system the line capacitances connected in parallel to ZT0 and 3ZE can be neglected due to the low values of ZE. A fault current with considerable magnitude will flow. For an isolated network, no current can flow through the transformer zero sequence impedance. This results in only a small earth leakage current flowing through the line capacitances. For a resonantly earthed system, the reactance of the Petersen coil connected in parallel with the line capacitances results in a resonant circuit which blocks the fault current for a single-phase to earth fault. The small amount of ‘rest’ current still flowing results from the copper looses of the inductances as well as the inductive and capacitive spill current resulting from an over- or under-compensation.
- 24 - a) ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 3 ZE CL/2CL/2 b) ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 3 ZE CL/2CL/2 c) ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 3 XE CL/2CL/2 1CL3 LE 2 0 =⋅⋅ω Figure 6.2: Single phase-to-earth short-circuit with different starpoint earthing arrangements: a) solid earthed starpoint b) isolated starpoint c) resonantly earthed starpoint (compensated system) The advantage of resonant earthing (or earth fault compensation) is the fact that in networks with "self-healing" isolation (e.g. overhead lines) earth faults extinguish automatically, as soon as the fault cause (e.g. lightning overvoltages or short circuits caused by trees in strong gusts of wind) is removed. In a compensated system also the earth fault rest-current is significantly smaller than the total capacitive earth fault circuit flowing in an isolated system. Due to the difficulty of extinguishing a capacitive current this is a significant advantage. A disadvantage of a compensated system, as with an isolated system, is that an earth fault on one phase results in a considerable voltage increase on the unfaulted / healthy phases. If insufficient isolation exists or if the insulation of a cable has been damaged previously, this will inevitably result in subsequent earth faults.
- 25 - 7 Appendix 7.1 References [1] H. Happoldt, D.Oeding: Elektrische Kraftwerke und Netze (5. Auflage). ISBN 3-540-08305-7, Springer Verlag Berlin 1978, [2] Oswald, Bernd: Netzberechung - Berechnung stationärer und quasistationärer Betriebszustände in Elektroenergieversorgungsnetzen. ISBN 3-8007-1718-2. VDE-Verlag, Berlin, 1992. [3] N.N.: Short circuit currents in three-phase A/C systems. Internationale Norm IEC 60909 -.1. Ausgabe 2001-07, International Electrotechnical Commission, Genf, Schweiz. [4] Balzer, G. et al.: Die Kurzschlussstromberechnung nach IEC 60909 - Unterschiede zwischen alter und neuer überarbeiteter Auflage. Bulletin des Schweizerischen Elektrotechnischen Vereins SEV Bulletin, 18. Januar 2002, 93. Jahrgang, Zürich, Schweiz. [5] N.N.: DIgSILENT PowerFactory, Technical Reference Manual 7.2 Symbols i Current as (time-dependent) amplitude value U Voltage I Current as rms value S Apparent power c Voltage correction factor according to IEC60909 K Impedance correction factor according to IEC60909 t Ratio of the transformer ϕ Angle between voltage and current p Deviation of the operating voltage from the setpoint voltage in p.u. (for generators and transformers) κ Auxiliary factor for IEC 60909 to consider the maximum asymmetric short-circuit current λ Auxiliary factor for IEC 60909 to consider the breaking current for synchronous machines µ Auxiliary factor for IEC 60909 to consider the breaking current for asynchronous machines R Resistance (real component of impedance) X Reactance (reactive component of impedance) α Temperature coefficient of the resistivity (material constant) ϑ Temperature (in °C) 7.3 Hyphens " Sub-transient value ' Transient value Steady state value
- 26 - 7.4 Indices L Line (cable / overhead line) N External grid T Transformer R Reactance / reactor / current limiting coil S Synchronous machine G Generator (in IEC 909/60909 usually equivalent to synchronous machine) A Asynchronous machine M Motor (in IEC 909/60909 usually equivalent to asynchronous machine) PS Power station E Earthing system Load Load Conv Converter / inverter 1,2,0 Magnitude / value in positive, negative and zero sequence system LR Condition of the asynchronous machine with locked rotor AC Alternating current component DC Direct current component k Short-circuit Σij Impedance from side i to side j with impedances switched in series n Nominal value r Rated value d (q) Values on the d (q) axis Op Operating value b Operating value (according to IEC60909) min Calculation of the minimum short-circuit current max Calculation of the maximum short-circuit current HV High voltage MV Medium voltage LV Low voltage

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DigSILENT PF - 05 short circuit theory

  • 1. Update: June 12, 2003 Short-Circuit Calculations Basic Principles and Models Training Course Documents
  • 2. Contents 1 The Role of Short-Circuit Calculations 1 1.1 Areas of Applications of Short-Circuit Calculations 1 1.2 Time Dependence of Short-Circuit Current 2 1.3 The System of the Symmetrical Components 3 1.4 Short-Circuit Classification According to Involved Phases 5 2 Plant Models 7 2.1 External Grid 7 2.2 Overhead Lines and Cables 7 2.3 Two-Winding Transformer 8 2.4 Three-Winding Transformer 8 2.5 Series Reactance (Short-Circuit Current Limiting Reactor) 9 2.6 Synchronous Machine 9 2.7 Asynchronous Machine 10 2.8 Loads and Static Shunt-Compensators 10 3 Superposition Method for Short-circuit Calculations 11 4 IEC 60909 Method 12 4.1 Derivation of the Method 12 4.2 Correction Factors of IEC 60909 13 4.3 The Changes from IEC909: 1988 to IEC 60909:2001 15 5 Short-Circuit Currents in the Different Time Domains (according to IEC) 17 5.1 Classification of the Source of Short-Circuits 17 5.2 Initial Symmetrical Short-Circuit Current I"k (according to IEC 60909) 18 5.3 Peak Short-Circuit Current (according to IEC 60909) 18 5.4 Decaying (Aperiodic) DC Component of the Short-Circuit Current idc 20 5.5 Symmetrical Short-Circuit Breaking Current Ib 21 5.6 Steady-State Short-Circuit Current Ik 22 5.7 Thermal Equivalent Short-Circuit Current Ith 22 6 Earthing of Distribution Networks 23 7 Appendix 25 7.1 References 25 7.2 Symbols 25 7.3 Hyphens 25 7.4 Indices 26
  • 3. - 1 - 1 The Role of Short-Circuit Calculations 1.1 Areas of Applications of Short-Circuit Calculations Apart from the load flow calculation, short-circuit analysis is the most frequently used calculation function when dealing with electrical networks. It is used in system planning as well as system operations (see Figure 1.1). Figure 1.1: Areas of application for short-circuit calculations [2] Applications in system planning are for example: • Ensuring that the defined short-circuit capacity of equipment is not exceeded with system expansion and system strengthening. • Co-ordination of protective equipment (Fuses, over-current and distance relays). • Dimensioning of earth mats • Verification of sufficient fault level capacities at load points (e.g. uneven loads like arc furnaces, thyristor-driven variable speed drives or dispersed generation. • Verification of allowed thermal limits of cables and transmission lines. Applications in system operations are for example: • Ensuring that short-circuit limits are not exceeded when changing the system configuration • Determining protective relay settings as well as fuse sizing • Calculation of fault location for protective relays, which store fault disturbance recordings. • Analysis of system faults, e.g. mal-operation of protection equipment. • Analysis of possible mutual interference of parallel lines during system faults. The fundamental difference for the calculation assumptions is that for system planning studies the system operating conditions are not yet known, and therefore estimates are necessary. For this purpose the method of the equivalent voltage source at the fault location has generally become accepted in Western Europe according to IEC 909 (VDE 0102). A revised version of this was published as IEC 60909 in July 2001. This method works independently to the loadflow of a system. It is based on the nominal and/or calculated dimensions of the operating plant of a system and uses correction factors for voltages and impedances, to ‘push’ the results towards the safe side. For the calculation of minimum and maximum short-circuit currents, different correction factors are applied. Operating / Working Conditions Online Short-Circuit Calculations Planning Criteria Simplified procedure (IEC, ANSI, ...) Reduced Data Set Detailed procedures Complete Data Set Method 1: Equivalent Voltage Source at the Fault Location Method 2.1: Beat Method / Superposition Method Method 2.2: Lösung der DGL sub-transient (initial symmetrical) Short-circuit current I k " ip Ib Ith κ µ m, n I"k, Uk i ik(t) Operating Conditions Online short -circuit calculations Planning Conditions Simplified method (IEC, ANSI, ...) Reduced data set Detailed method Complete data set Method 1: Equivalent voltage source at fault location Method 2.1: Superposition method Method 2.2: Solving of differential eq. Initial symmetrical (subtransient) short- circuit current I k " ip Ib Ith κ µ m, n I"k, Uk i ik(t)
  • 4. - 2 - For short-circuit calculations in a system operation environment the exact network operating conditions are well known. If the accuracy of the calculation according to IEC 60909 is not sufficient - or to verify the results of this method, the superposition method can be used. It calculates the expected short-circuit currents in the network on the basis of the existing network operating condition. If the system models are correct, the results from this method are always more exact than the results of the method according to IEC 60909. The system analyst is, however, responsible that he has chosen the most unfavourable conditions with respect to the sizing of plant. In individual cases, this might result in extensive studies required. 1.2 Time Dependence of Short-Circuit Current The time dependence of short-circuit current is of importance with respect to the loading experienced by affected plant. In principle one distinguishes between a short circuit that is ‘near to’ or ‘far from’ a generator. (Figure 1.2). Top envelope DC-Component idc Time Current Bottom envelope Top envelope DC-Component idc Time Current Bottom envelope a) Far from generator b) Near to generator Figure 1.2: Time dependence of short-circuit current The distinction of the time dependence is due to the effect of the higher stator current (due to the close fault) on the induced currents in the damper windings, rotor mass and field winding. In the case of a fault near to a generator the stator current can increase so much that the resulting magnetic field weakens the rotor field considerably. As a consequence the terminal voltage collapses. The associated positive sequence model of a synchronous machine is shown in Figure 1.3. The delayed effect of the stator field on the excitation (rotor) field is modelled by switching between the source voltage E", E' and E depending on the time frame of the calculation. xd-x'd S x'd-x"d x"d E' E"E ∞ t Figure 1.3: Single-phase equivalent circuit diagram of a generator for short-circuit current calculations which include the modelling of the field attenuation For all practical calculations the IEC60909 standard assumes a fault near to a generator if the fault current in at least one generator exceeds twice the rated current. The short-circuit current is described by the following parameters: ip Peak short-circuit current, i.e. magnitude of the first instantaneous peak value of the short-circuit current. It is used for the calculation of the mechanical load of electrical plant (e.g. busbars in switchgears, transformers). This value depends on the R/X relationship of the fault impedance and the angle of the short-circuit current inception. idc Decaying (aperiodic / direct-current) component of the short-circuit current. This value is of interest to determine the thermal effect of the short-circuit current. In the case of faults near to the generator, it also affects the time up to the first zero crossing of the current.
  • 5. - 3 - Ik" Initial symmetrical (subtransient) short-circuit current. This is the rms value of the alternating current at the inception of the short-circuit. Ik' Transient short-circuit currentThis is the rms value of the alternating current at the transition from sub-transient short-circuit current to steady state short-circuit current. Ik Steady state short-circuit current. This is the rms value of the alternating current in the steady-state condition. Not possible to determine from the time dependence, but often used as a derived quantity: knk "IU3"S ⋅⋅= Initial symmetrical (subtransient) short-circuit power. This is calculated as the fictitious product of the rated voltage Un (approximated as the voltage before short-circuit inception) and the initial short- circuit current Ik". 1.3 The System of the Symmetrical Components 1.3.1 Concept of Symmetrical Components Using the method of the symmetrical components a three-phase AC system (including the capacitive and inductive couplings) can be split into three independent single phase systems namely Positive, Negative and Zero sequence networks. Three phasors are introduced, which have a magnitude of unity, but are 120° phase displaced: 1, a and a2 . With: 2 3 j 2 1 a +−= (Eq. 1.1) 2 3 j 2 1 a2 −−= (Eq. 1.2) 0aa1 2 =++ (Eq. 1.3) Im Re 1 = a3 a2 a 120° 120° 120° Figure 1.4: Symmetrical system with phasors 1, a and a2 a) I1R ωt I1S I1T b) I1R ωt I1T I1S c) I1R ωt I1T I1S Figure 1.5: Example for Positive, Negative and Zero Sequence currents
  • 6. - 4 -           ⋅           ⋅=           T S R I I I aa aa I I I 2 2 2 1 0 1 1 111 3 1 (Eq. 1.4) i.e. ( )TSR IIII ++= 3 1 0 ( )TSR IaIaII ⋅+⋅+= 2 1 3 1 ( )TSR IaIaII ⋅+⋅+= 2 2 3 1           ⋅           =           2 1 0 2 2 1 1 111 I I I aa aa I I I T S R (Eq. 1.5) i.e. 210 IIII R ++= 21 2 0 IaIaII S ⋅+⋅+= 2 2 10 IaIaII T ⋅+⋅+= 1.3.2 Determining the Sequence Impedance Values The impedance of electrical plant in the system of the symmetrical components can be determined by switching a voltage supply of the appropriate phase sequence to the terminals of the plant and measuring the resulting currents. The equivalent circuits are shown below. a) b) Figure 1.6: Equivalent circuit to determine the positive sequence impedance (a) and negative sequence impedance (b) Figure 1.7: Equivalent circuit to determine the zero sequence impedance
  • 7. - 5 - 1.4 Short-Circuit Classification According to Involved Phases Depending on the phases which are involved in a short-circuit, one needs to distinguish between a three-phase short circuit, a phase-phase short circuit, a phase-phase-earth short-circuit or a single phase-earth short-circuit. The different types of short-circuit are shown in Figure 1.8 to 1.11. The illustrations show the system fault in the ABC phase representation (above) and the equivalent circuit in the system of the symmetrical components (below). 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 nU c ⋅ 1I Figure 1.8: Three-phase short-circuit (ABC phase representation and equivalent circuit in symmetrical components) 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 U c n ⋅ 1I 2I Figure 1.9: Phase-to-phase short-circuit (no earth connection) (ABC phase representation and equivalent circuit in symmetrical components)
  • 8. - 6 - 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 nU c ⋅ 1I 2I 0I Figure 1.10: Phase-phase-earth short-circuit (ABC phase representation and equivalent circuit in symmetrical components) 1 2 0 L1 L2 L3 ZA1 ZA2 ZA0 ZB1 ZB2 ZB0 ~3 nU c ⋅ 1I 0I 2I Figure 1.11: Single-phase to earth fault (ABC phase representation and equivalent circuit in symmetrical components) The designation earth contact and earth short-circuit depends on whether the currents flowing during an earth fault are short-circuit or load current like. This depends, of course, on the system earthing. One speaks of an earth contact in isolated or resonantly earthed networks, and of earth short-circuits in solidly earthed or resistively earthed systems. In order to calculate the short-circuit current of a network, the positive and negative sequence data for the plant as well as the zero sequence data is required.
  • 9. - 7 - 2 Element Models For the common element models, the calculated (or actual measured) values are used. The shown models are not complete as required, for example, for loadflow calculation. They rather represent simplifications, which are made in the context of the short-circuit calculations according to the IEC. The detailed equivalent circuit diagrams can be viewed in the PowerFactory manual [5]. 2.1 External Grid ~U11 2 0 RN1 XN1 RN2 XN2 RN0 XN0 Figure 2.1: Short-circuit model for external grid Parameters and calculations: k n 1N "I3 Uc Z ⋅ ⋅ = (Eq. 2.1) Additional data: 1N2N ZZ = RN1, XN1 according to Ratio 1N 1N X R RN0, XN0 according to 1N 0N Z Z und 0N 0N X R 2.2 Overhead Lines and Cables 1 2 0 RL1 XL1 RL2 XL2 RL0 XL0 CL0/2 CL0/2 Figure 2.2: Short-circuit model of lines Parameters and calculations: RL1, XL1 according to conductor geometry or manufacturer data 12 LL ZZ = RL0, XL0 according to conductor/geometry under consideration of additional parallel conductors buried in the ground (e.g. district-heating tubes). Thermal resistance compensation for the calculation of the minimum short-circuit current ( )[ ] 20,LeL RC201R ⋅°−ϑ⋅α+= (Eq. 2.2)
  • 10. - 8 - 2.3 Two-Winding Transformer 1 2 0 RT,HV1 XT,HV1 RT,HV2 XT,HV2 3ZE1 3ZE2 ZT0 Figure 2.3: Short-circuit model for a two-winding transformer Parameters and calculations: rT 2 HV,rT kr1HV,T S U uZ ⋅= (Eq. 2.3) rT 2 HV,rT Rr1HV,T S U uR ⋅= (Eq. 2.4) 1HV,T2HV,T ZZ = (Eq. 2.5) The zero-sequence equivalent circuit is dependant on the vector group of the transformer (see users manual [5]) 2.4 Three-Winding Transformer 1 0 RT,HV11 XT,HV11 3ZE1 3ZE3 ZT0 RT,HV31 XT,HV21RT,HV21 XT,HV31 2 3ZE2 RT,HV12 XT,HV11 RT,HV32 XT,HV22 RT,HV22 XT,HV32 Figure 2.4: Short-circuit model for a Three-winding transformer Parameters and calculations: 12rT 2 HV,rT 12kr1,12 S U uZ ⋅=Σ (Eq. 2.6) 12rT 2 HV,rT 12Rr1,12 S U uR ⋅=Σ (Eq. 2.7) 23rT 2 HV,rT 23kr1,23 S U uZ ⋅= ∆Σ (Eq. 2.8) 23rT 2 HV,rT 23Rr1,23 S U uR ⋅=Σ (Eq. 2.9) 31rT 2 HV,rT 31kr1,31 S U uZ ⋅=Σ (Eq. 2.10) 31rT 2 HV,rT 31Rr1,31 S U uR ⋅=Σ (Eq. 2.11) ( )1,311,231,1211HV,T ZZZ 2 1 Z ΣΣΣ +−= ( )1,311,231,1221HV,T ZZZ 2 1 Z ΣΣΣ −+= (Eq. 2.12) ( )1,311,231,1231HV,T ZZZ 2 1 Z ΣΣΣ ++−= (Eq. 2.13) 1,HVi,T2,HVi,T ZZ = The zero-sequence equivalent circuit is dependant on the vector group of the transformer (see users manual [5])
  • 11. - 9 - 2.5 Series Reactance (Short-Circuit Current Limiting Reactor) 1 2 0 RR1 XR1 RR2 XR2 RR0 XR0 Figure 2.5: Short-circuit model for a series reactance Parameters and calculations: rR 2 n kr1R I3 U uZ ⋅ ⋅= (Eq. 2.14) rT 2 rT Rr1R S U uR ⋅= (Eq. 2.15) For a balanced system: 1R0R2R ZZZ == (Eq. 2.16) 2.6 Synchronous Machine ~U"11 2 0 RS1 X"S1 RS2 X"S2 RS0 X"S0 ZE 3ZE S Figure 2.6: Short-circuit model for a synchronous machine Parameters and calculations: dSS "jXRZ += (Eq. 2.17) Additional data: 2S2S2S jXR"jXRZ +=+= (Eq. 2.18) Normally it is assumed that X2 = x"d . If x"d and xq" differ significantly in magnitude, the following can be used: ( )qd xxXX "" 2 1 " 22 +⋅== (Eq. 2.19)
  • 12. - 10 - 2.7 Asynchronous Machine ASM ~U"11 2 0 RA1 X"A1 RA2 X"A2 RA0 X"A0 Figure 2.7: Short-circuit model for an Asynchronous machine Parameters and calculations: rM 2 rM rM LR AK S U I I 1 Z ⋅       = (Eq. 2.20) 2.8 Loads and Static Shunt-Compensators 1 2 0 RLoad1 XLoad1 not for IEC60909 CLoad1 RLoad2 XLoad2 CLoad2 RLoad0 XLoad0 CLoad0 0 Figure 2.8: Short-circuit model for loads and shunt – compensators The complete equivalent circuit for loads is only used for the superposition method. If IEC60909 is used, the loads are not reflected in the positive and negative sequence networks, but are of importance in the zero-sequence network.
  • 13. - 11 - 3 Superposition Method for Short-circuit Calculations The superposition method is (in terms of system modelling) an accurate calculation method. The fault currents of the short-circuit are determined by overlaying the healthy loadflow condition before short-circuit inception with a condition where all voltage supplies are set to zero and the negative operating voltage is connected at the fault location. The procedure is shown in Figure 3.1 below. ~ ~ ~ US1 US2 US3 UOp,0 ~ UOp,0 + = UOp,0 ~ ~ ~ US1 US2 US3 USC= 0 IOp IOp IOp ISC ISC ISC ISC + IOp ISC + IOp ISC + IOp a) b) c) Figure 3.1: Superposition method for short-circuit calculations Starting point is the operating condition of the system before short-circuit inception (see Figure 3.1 a). This condition represents the excitation conditions of the generators, the tap positions of regulated transformers and the breaker / switching status of the operational plant. From this pre-fault condition the pre-fault busbar voltage of the faulted busbar can be calculated. For the pure fault condition the system condition is calculated for the situation where, the negative pre-fault busbar voltage for the faulted bus is connected at the fault location and all other sources / generators are set to zero (see Figure 3.1 b). Since network impedances are assumed to be linear, the system condition after fault inception can be determined by overlaying (complex adding) both the pre-fault and pure fault conditions (se Figure 3.1 c).
  • 14. - 12 - 4 IEC 60909 Method 4.1 Derivation of the Method The method of the equivalent voltage source at the faulted bus is a simplification of the superposition method with the goal of accomplishing a close-to-reality short-circuit calculation without the need for the preceding load flow calculation and the associated definition of actual operating conditions. Figure 4.1 shows, how the method of the equivalent voltage source can be derived from the superposition method. ~ ~ ~ US3=Un3 UOp,0=Un ~ + Un ~ ~ ~ Un1 Un2 Un3 USC= 0 IOp=0 ISC ISC ISC ISC ISC ISC a) b) c) US2=Un2 US1=Un1 IOp=0 IOp=0 c Un ≈ Figure 4.1: Method of the equivalent voltage source at the faulted busbar, derived from the superposition method In comparison, the main simplifications in comparison to the superposition method are the following: • Nominal conditions are assumed for the whole network, i.e. Ui = Un,i. • Load currents are neglected, i.e. IOp,i = 0. • A simplified simulation network is used, i.e. loads are not considered in the positive and negative sequence network. • To ensure that the results are estimated on the safe side, a correction factor c is applied to the voltage at the faulted busbar. This factor differs for the calculation of the maximum and the minimum short-circuit current of a network.
  • 15. - 13 - 4.2 Correction Factors of IEC 60909 The superposition method always assumes a realistic system condition as the basis for a short-circuit calculation, which is determined from a load flow calculation preceding the short-circuit calculation. Thus the exact short-circuit currents are determined for this system condition. By using the equivalent voltage source in accordance with IEC60909, the aim is to calculate the maximum and minimum short-circuit currents for all possible operating conditions with only one calculation. Only the rated voltage of the faulted bus is required. For this purpose IEC 60909 introduces a voltage correction factor and several impedance correction factors. The concept of impedance correction according to IEC 60909 (see Figure 4.2) is to correct the source impedance in such a way, that in the case of calculating the short-circuit current of a circuit with the voltage c×Un and corrected impedance the same values results as in the case of calculating with the actual fault voltage and actual impedance: ~U"k Zk I"k ~c Un K Zk I"k,IEC Figure 4.2: Concept of impedance correction according to IEC 60909 Thereby the following is applicable: ( ) IEC,k k n k k k "I ZK3 Uc Z3 "U "I = ⋅⋅ ⋅ ≡ ⋅ = (Eq. 4.1) The values for voltage factor c as well as the impedance correction factors K are listed below. 4.2.1 Voltage Factor Table 4.1: Voltage factor c as a function of the nominal voltage Rated / nominal voltage Calculation of max. short- circuit current cmax Calculation of min. short-circuit current Cmin Low Voltage Un ≤ 1 kV 1.05 (with Umax ≤ 1.06 Un) 1.10 (with Umax ≤ 1.10 Un) 0.95 Medium Voltage 1 kV < Un ≤ 35 kV 1.10 1.00 High Voltage 35 kV < Un 1.10 If Un is not defined: cmax⋅Un → Um 1.00 If Un is not defined: cmin⋅Un → 0.9⋅Um In general ensure: cmax ⋅ Un ≤ Um 4.2.2 Impedance Correction for Power Transformers Correction of the transformer impedances in the positive, negative and zero sequence networks (except for earth impedances): krT max T x6.01 c 95.0K ⋅+ ⋅= (Eq. 4.2)
  • 16. - 14 - If the operating conditions of the transformer prior to the fault inception are known, the following correction factor can be used: max,Tb rT max,Tb rT max max,b n T sin I I x1 c U U K ϕ      ⋅+ ⋅= (Eq. 4.3) Impedance correction in three-winding transformers: 12kr max 12T x6.01 c 95.0K Σ Σ ⋅+ ⋅= (Eq. 4.4) 23kr max 23T x6.01 c 95.0K Σ Σ ⋅+ ⋅= (Eq. 4.5) 31kr max 31T x6.01 c 95.0K Σ Σ ⋅+ ⋅= (Eq. 4.6) 4.2.3 Impedance Correction for Synchronous Machines (Generators) Using IEC 60909, the actual measured resistive portion of the short-circuit impedance Rs may not be used for the real portion of the short-circuit impedance. Instead a fictitious resistance value RSG is introduced, which is significantly higher in comparison to RS and should simulate the decaying DC component. The values of the resistance RSG to be used are shown below. Table 4.2: Choice of the fictitious generator resistance RS/X"d UrG SrG 0.15 ≤ 1kV arbitrary 0.07 > 1kV < 100 MVA 0.05 > 1kV ≥ 100 MVA In addition the positive, negative and zero sequence impedances (with exception of the earthing impedances) of the synchronous machine are to be corrected with the factor KG: rGd max rG n G sin"x1 c U U K ϕ⋅+ ⋅= (Eq. 4.7) 4.2.4 Impedance Correction for Power Stations 4.2.4.1 Power stations with on-load tap changers Impedance correction in the positive, negative and zero sequence network (with exception of the earthing impedances) for power stations: rGTd max 2 r 2 rG 2 Netw,n PS sinx"x1 c t 1 U U K ϕ⋅−+ ⋅⋅= (Eq. 4.8) If the power station is operating under abnormal conditions, (e.g. operating with a voltage at the generator terminals that deviates from UrG, under-excited operation), IEC60909 defines a number of corrections, which “should” be used instead. 4.2.4.2 Power stations with off-load tap changers Impedance correction in the positive, negative and zero sequence network (with exception of the earthing impedances) of the power station: ( ) ( ) rGd max T rGrG Netw,n PS sin"x1 c p1 t 1 p1U U K ϕ⋅+ ⋅+⋅⋅ +⋅ = (Eq. 4.9) If the power station is operating under abnormal conditions, (e.g. operating with a voltage at the generator terminals that deviates from UrG, under-excited operation), IEC60909 defines a number of corrections, which “should” be used instead.
  • 17. - 15 - 4.2.5 Guidelines for the Modelling of Asynchronous Machines For the real part of the short-circuit impedance - similar to the synchronous machine - reference values are indicated as a function of the rated voltage and power for each pole pair. These are shown in the table below. Table 4.3: Selection of the resistance (according to IEC 0909) RM/XM UrM PrM per pole pair 0.1 > 1kV ≥ 1 MW 0.15 > 1kV < 1 MW 0.42 ≤ 1kV, incl. connection cable arbitrary / random IEC 60909 furthermore defines a number of conditions, under which the fault contribution of asynchronous machines can be neglected. These assumptions are only of meaning in a manual short-circuit calculation. When using a computer simulation program, however, they can be neglected. Therefore reference is only made to section 3.8.2 in [3]. 4.3 The Changes from IEC909: 1988 to IEC 60909:2001 The superposition method is always based on a realistic network condition, which is determined by running a loadflow prior to the actual short-circuit calculation. Thus the exact short-circuit current for this system condition is calculated. The intention of the use of an equivalent voltage source according to IEC60909, is to calculate the maximum and the minimum short-circuit currents for all possible operating conditions with exactly one calculation. In addition only the nominal voltage is to be used at the fault location. The norm therefore introduces a voltage correction factor and several impedance correction factors. 4.3.1 Correction Factors for Power Transformers IEC 909:1988 No impedance correction in principle. Reference is made to special considerations for the following cases: - A single fed short-circuit has the same direction as the load current - Tap changer with an overall voltage ratio > 5% - Voltage Uk,min is significantly lower than the short-circuit voltage Ukr - Operating voltage is significantly higher than Un (U>1.05 Un) IEC60909: 2001 Specific correction factor for power transformers 4.3.2 Correction Factors for Synchronous Generators without Generator Transformers IEC909: 1988 Calculation of the impedance correction with UrG IEC 60909:2001 Substitution of ( ) rGGrG Up0.1U ⋅+→ , if the generator operating voltage continuously deviates from UrG
  • 18. - 16 - 4.3.3 Correction Factors for Power Stations 4.3.3.1 Factors for power stations with tap change controller IEC909: 1988 Choice between: - Single correction: Impedances of generator and step-up transformer are multiplied by different correction factors. - Total correction: Impedances of generator and step-up transformer are regarded as one impedance and are corrected together. IEC 60909:2001 Only total correction allowed 4.3.3.2 Factors for power stations without tap change controller IEC909: 1988 (No guidelines) IEC 60909:2001 Own correction factor 4.3.4 Maximum Cable Temperature for the Calculation of the Minimum Short-Circuit Current IEC 909:1988 ϑmax = 80 °C IEC60909: 2001 ϑmax corresponds to the maximum possible conductor temperature 4.3.5 Voltage Factor c for Low Voltage Systems 400 / 230V IEC 909:1988 c max = 1.0 IEC60909: 2001 c max = 1.05 (for Ub < 1.06 Un), otherwise cmax = 1.10 4.3.6 Consideration of the Fault Contribution of Asynchronous Machines IEC 909:1988 - consideration of three phase and two phase (to earth) faults - note that for low resistance earthing the single phase to earth fault is also to be considered (no calculation instruction given) IEC60909: 2001 - detailed calculation instruction for all types of fault - concrete rules for the consideration of asynchronous machines (meaningful only for manual calculations) 4.3.7 Standardisation of New Calculation Methods IEC60909: 2001 - calculation instruction for single phase interruption in the medium voltage system (fuse blowing) and faults in the LV system - calculation instruction for the thermal effective short-circuit current as adopted from IEC 865-1
  • 19. - 17 - 5 Short-Circuit Currents in the Different Time Domains (according to IEC) 5.1 Classification of the Source of Short-Circuits The different types of sources are treated differently during short-circuit current calculations according to IEC 60909. Depending on the type of source, some of the characteristic current values derived from the initial symmetrical short- circuit current I"k are calculated differently. For a single fed source (Figure 5.1) there exists only one path from the feeding voltage source to the fault location. Depending upon the number of such voltage sources one distinguishes between a single source (exactly one voltage source) and a non-meshed source (several voltage sources). For non-meshed sources the portion of short-circuit current that flows in each branch of the network is the current supplied by the one and only source connected to this branch. A meshed source exists (Figure 5.2) if the short-circuit lies in a part of the network, where the portion of short-circuit current in at least one branch of the network is supplied from more than one source. a) b) Figure 5.1: Single fed short-circuits fed from a single source (a) and a Non-meshed network (b) Figure 5.2: Short-circuit is a meshed network
  • 20. - 18 - 5.2 Initial Symmetrical Short-Circuit Current I"k (according to IEC 60909) 5.2.1 Maximum Initial Symmetrical Short-Circuit Current I"k,max The following steps are performed: - Reproduce the system in symmetrical components, including the indicated impedance corrections. - Insert the equivalent voltage source at the fault location and the calculate k 2 nmax max,k Z 1 3 Uc "I ⋅ ⋅ = (Eq. 5.1) - The currents in the three-phase system are derived from the inverse transformation of the symmetrical components currents. 5.2.2 Minimum Initial Symmetrical Short-Circuit Current I"k,min The procedure is basically the same as the calculation of the maximum initial symmetrical short-circuit current I"k,max. The following differences exist [3]: - The voltage correction is done with the factor cmin instead of cmax. - For overhead lines and cables the resistance values for the maximum possible conductor temperature (instead of 20ºC) should be specified. - The system and supplying generators are to be configured such that the lowest possible short circuit current results. 5.3 Peak Short-Circuit Current (according to IEC 60909) The following steps are performed: - Reproduce the system in symmetrical components, including the indicated impedance corrections. - Insert the equivalent voltage source at the fault location and the calculate kp "I2i ⋅⋅κ= (Eq. 5.2) The calculation of κ is dependant on whether the source is single or meshed. 5.3.1 Calculation of the Factor κ for Unmeshed Systems The value of κ is calculated from the short-circuit impedance Zk = Rk + j Xk as follows: k k X R 3 k k e98.002.1 X R ⋅− ⋅+=      κ=κ (Eq. 5.3) The factor κ is used uniformly for all types of fault.
  • 21. - 19 - Figure 5.3: Dependence of κ with respect to R/X ratio 5.3.2 Calculation of the Factor κ for Meshed Systems If the decaying direct current component is determined from branches with differing R/X ratios, then the factor κ must be calculated using one of the following methods. The method of equivalent frequency is generally used for computer-aided calculations, as it is simple to implement, yet results in relative high accuracy. 5.3.2.1 Uniform R/X Ratio (Method A) For this method the minimum R/X ratio of all branches through which any portion of the short-circuit current flows is selected. Network plant connected in series will be considered as one branch.       =      i i min X R Min X R i: branches carrying a portion of the short-circuit current (Eq. 5.4)               κ=κ min a X R (Eq. 5.5) This procedure results is a conservative estimate for the value of R/X and therefore leads to larger κ values than in reality. 5.3.2.2 R/X Ratio at the Fault Location (Method B) Here the R/X ratio is determined according to the ratio of the impedance components at the fault location. This generally results in a too small value of κ. Therefore the calculated peak short circuit current needs to be corrected.       κ=κ k k b X R (Eq. 5.6) kbp Ii "215.1 ⋅⋅⋅= κ (Eq. 5.7) The following special rules need to be considered a) If R/X is less than 0.3 for all branches, the 1.15 correction factor need not be applied. b) For low-voltage systems the value of 1.15 x κb should not exceed 1.8. c) For medium / high-voltage transmission systems the value of 1.15 x κb should not exceed 2.0.
  • 22. - 20 - 5.3.2.3 The Equivalent Frequency Method to Determine the R/X Ratio (Method C) This method consists of three steps: - Adjust the resistances and reactances for all plant components according to Eq. 5.8 and 5.9. The value to use for the equivalent frequency fe is listed in Table 5.1. i * i RR = (Eq. 5.8) e n i * i f f XX ⋅= (Eq. 5.9) - Determine the short-circuit impedance at the fault location for the equivalent frequency fe. - The value of κc is determined from the ratio of the short-circuit impedance components at the equivalent frequency:         κ=κ e e f,k f,k c X R (Eq. 5.10) Table 5.1: Equivalent frequencies for 50 / 60 Hz systems. fn fe 50 Hz 20 Hz 60 Hz 24 Hz 5.4 Decaying (Aperiodic) DC Component of the Short-Circuit Current idc IEC 60909 suggests the following approximation for the time response of the decaying DC component. t X R f2 kdc e"I2i ⋅π− ⋅⋅= (Eq. 5.11) The R/X ratio is determined according to the method of uniform R/X ratio (5.3.2.1) or according to the method of equivalent frequency (5.3.2.3). Note that for synchronous machines the correct values of RS should be used and not the resistances RSf, which are used to determine the time constants of the DC components. For the method of equivalent frequency the equivalent frequency should be chosen according to the following table depending on the time frame studied. Table 5.2: Equivalent frequency fe to use to determine idc for different time frames. f⋅t <1 <2.5 <5 <12.5 fe/ fn 0.27 0.15 0.092 0.055
  • 23. - 21 - 5.5 Symmetrical Short-Circuit Breaking Current Ib 5.5.1 Unsymmetrical Faults Far from the Generator kb "II = (Eq. 5.12) 5.5.2 Symmetrical Faults Near to the Generator (single source) 5.5.2.1 Fault Contribution from Synchronous Machine / Generator kb "II ⋅µ= (Eq. 5.13) The factor µ(tmin) < 1 depends on the minimum breaker tripping time tmin. It allows for the feedback of the rotor field onto the stator. 5.5.2.2 Fault Contribution from Asynchronous Machine kb "IqI ⋅⋅µ= (Eq. 5.14) The additional factor q (tmin) < 1 depends on the minimum breaker tripping time tmin as well as the rated power per pole pair of the machine. It allows for the decay of the short circuit current contribution of the machine during solid short circuits. 5.5.3 Symmetrical Faults for Meshed Networks 5.5.3.1 Multiple non-meshed sources ∑= i i,bb II (Eq. 5.15) 5.5.3.2 Meshed Sources The total symmetrical short-circuit breaking current can be approximated by adding the breaking currents of each individual source. In addition IEC 60909 provides a formula to improve the accuracy of the result.
  • 24. - 22 - 5.6 Steady-State Short-Circuit Current Ik 5.6.1 Unsymmetrical Faults Far from the Generator kk "II = (Eq. 5.16) 5.6.2 Symmetrical Faults Near to the Generator (Single Source) S,rmaxmax,k II ⋅λ= (Eq. 5.17) S,rminmin,k II ⋅λ= (Eq. 5.18) The factor λ depends on Xd,sat. IkG"/IrG < 1 depends on the minimum breaker tripping time tmin. It allows for the feedback of the rotor field onto the stator. 5.6.3 Symmetrical Faults Near to the Generator (Unmeshed Source) ∑= i i,kk II (Eq. 5.19) 5.6.4 Symmetrical Faults Near to the Generator (Meshed Source) kk "II = (Eq. 5.20) This rough approximation does not always yield satisfactory results. As an alternative, the short circuit current contributions can be approximated by determining the breaking current Ib (according to IEC 60909) with maximum breaker tripping time. This IEC 60909 compatible solution is implemented in PowerFactory as an option. 5.7 Thermal Equivalent Short-Circuit Current Ith This current is used to verify the thermal capabilities of cables based on the response of the power system protection. ( ) k 2 thk 2 k T 0 2 TITnm"Idti k ⋅=⋅+⋅=∫ (Eq. 5.21) k Tk 0 2 th T dti I ∫ = (Eq. 5.22) m: thermal contribution of the DC component n: thermal contribution of the AC component
  • 25. - 23 - 6 Earthing of Distribution Networks In this chapter the effect of network earthing on the fault current for a single-phase to earth short circuit is discussed. This application is particularly interesting as it allows, apart from the application for short circuit calculation methods, a general discussion about the different possibilities of earthing in a network. In Figure 6.1 the system to be studied is shown as a single line diagram as well as in the system of symmetrical components. The system model consists of an external grid of the HV transmission system, of a HV-MV step-down transformer with the possibility to earth the starpoint on the MV side via different earth impedances and a line, which represent the MV network. At the end of the line a single-phase to earth fault is placed. For a single-phase to earth fault the positive, negative and zero sequence networks have to be switched in series. The model of the Dy transformer in the zero sequence network exists of an isolated HV side, with the MV shorted to earth via ZT0 and 3ZE. ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 ZE 3 ZE 1PE CL/2CL/2 Figure 6.1: Single phase-to-earth short-circuit represented as a single line diagram and an equivalent symmetrical components network Figure 6.2 shows the equivalent circuit diagrams for a system with a solidly earthed starpoint (or a low resistance current limiting starpoint) (Figure 6.2a), for an isolated starpoint (Figure 6.2b) and for a resonantly earthed (compensated) system (Figure 6.2c). In the case of the solidly earthed system the line capacitances connected in parallel to ZT0 and 3ZE can be neglected due to the low values of ZE. A fault current with considerable magnitude will flow. For an isolated network, no current can flow through the transformer zero sequence impedance. This results in only a small earth leakage current flowing through the line capacitances. For a resonantly earthed system, the reactance of the Petersen coil connected in parallel with the line capacitances results in a resonant circuit which blocks the fault current for a single-phase to earth fault. The small amount of ‘rest’ current still flowing results from the copper looses of the inductances as well as the inductive and capacitive spill current resulting from an over- or under-compensation.
  • 26. - 24 - a) ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 3 ZE CL/2CL/2 b) ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 3 ZE CL/2CL/2 c) ~U11 2 0 ZN1 ZT1 ZL1 ZN2 ZT2 ZL2 ZN0 ZT0 ZL0 3 XE CL/2CL/2 1CL3 LE 2 0 =⋅⋅ω Figure 6.2: Single phase-to-earth short-circuit with different starpoint earthing arrangements: a) solid earthed starpoint b) isolated starpoint c) resonantly earthed starpoint (compensated system) The advantage of resonant earthing (or earth fault compensation) is the fact that in networks with "self-healing" isolation (e.g. overhead lines) earth faults extinguish automatically, as soon as the fault cause (e.g. lightning overvoltages or short circuits caused by trees in strong gusts of wind) is removed. In a compensated system also the earth fault rest-current is significantly smaller than the total capacitive earth fault circuit flowing in an isolated system. Due to the difficulty of extinguishing a capacitive current this is a significant advantage. A disadvantage of a compensated system, as with an isolated system, is that an earth fault on one phase results in a considerable voltage increase on the unfaulted / healthy phases. If insufficient isolation exists or if the insulation of a cable has been damaged previously, this will inevitably result in subsequent earth faults.
  • 27. - 25 - 7 Appendix 7.1 References [1] H. Happoldt, D.Oeding: Elektrische Kraftwerke und Netze (5. Auflage). ISBN 3-540-08305-7, Springer Verlag Berlin 1978, [2] Oswald, Bernd: Netzberechung - Berechnung stationärer und quasistationärer Betriebszustände in Elektroenergieversorgungsnetzen. ISBN 3-8007-1718-2. VDE-Verlag, Berlin, 1992. [3] N.N.: Short circuit currents in three-phase A/C systems. Internationale Norm IEC 60909 -.1. Ausgabe 2001-07, International Electrotechnical Commission, Genf, Schweiz. [4] Balzer, G. et al.: Die Kurzschlussstromberechnung nach IEC 60909 - Unterschiede zwischen alter und neuer überarbeiteter Auflage. Bulletin des Schweizerischen Elektrotechnischen Vereins SEV Bulletin, 18. Januar 2002, 93. Jahrgang, Zürich, Schweiz. [5] N.N.: DIgSILENT PowerFactory, Technical Reference Manual 7.2 Symbols i Current as (time-dependent) amplitude value U Voltage I Current as rms value S Apparent power c Voltage correction factor according to IEC60909 K Impedance correction factor according to IEC60909 t Ratio of the transformer ϕ Angle between voltage and current p Deviation of the operating voltage from the setpoint voltage in p.u. (for generators and transformers) κ Auxiliary factor for IEC 60909 to consider the maximum asymmetric short-circuit current λ Auxiliary factor for IEC 60909 to consider the breaking current for synchronous machines µ Auxiliary factor for IEC 60909 to consider the breaking current for asynchronous machines R Resistance (real component of impedance) X Reactance (reactive component of impedance) α Temperature coefficient of the resistivity (material constant) ϑ Temperature (in °C) 7.3 Hyphens " Sub-transient value ' Transient value Steady state value
  • 28. - 26 - 7.4 Indices L Line (cable / overhead line) N External grid T Transformer R Reactance / reactor / current limiting coil S Synchronous machine G Generator (in IEC 909/60909 usually equivalent to synchronous machine) A Asynchronous machine M Motor (in IEC 909/60909 usually equivalent to asynchronous machine) PS Power station E Earthing system Load Load Conv Converter / inverter 1,2,0 Magnitude / value in positive, negative and zero sequence system LR Condition of the asynchronous machine with locked rotor AC Alternating current component DC Direct current component k Short-circuit Σij Impedance from side i to side j with impedances switched in series n Nominal value r Rated value d (q) Values on the d (q) axis Op Operating value b Operating value (according to IEC60909) min Calculation of the minimum short-circuit current max Calculation of the maximum short-circuit current HV High voltage MV Medium voltage LV Low voltage